From the equation, an increase in G will increase Y, by the
amount of increase in G multiplied by the fiscal multiplier for the planned
policy. This will then lead to an increase in T, thus the Gt – Tt (the budget
deficit) is not increased by 1 X ∆G but ∆G - ∆Tt, where:

∆G = change in government expenditure as a result of the fiscal
stimulus

∆Tt = change in tax receipt as a result of the fiscal stimulus

As mentioned in this post and the previous
post, the fiscal multiplier is very important in determining the
effectiveness of government spending and this fiscal multiplier depends on a
lot of factors such as the design of the stimulus (the spending components
involved), the structure of the economy, and consumer and business sentiments.
In US and Australia, where infrastructures lags behind a lot of countries such
as South Korea, Japan and China, spending on infrastructures have a very large
multiplier. From this
post by Noah Smith, it showed a table from a NBER working paper by Alan Auerbach and Yuriy Gorodnichenko
estimating the effect of government expenditure on consumption and investment
for the US:

The important thing to take note here is that although investment
spending has a larger multiplier than consumption spending, it has less
immediate effects on the economy. Thus if the government wishes to stabilise
the economy (stabilising the lost in income due to increase in unemployment)
immediately when the economy faces a downturn, consumption spending works
better but it does not necessarily “pay for itself”[1] in the long run, while
investment spending (such as building roads, railways, investing in education,
health) provides longer term benefits such as productivity gains, reducing
unemployment in the longer run, and it is very likely to “pay for itself”.

Consumption spending also has one other very important effect
which investment spending lacks; this effect is the stabilising of consumer and
business sentiments in a downturn. The earlier and quicker the stimulus is
enacted when an economy faces downturn, the less consumer and business
sentiment falls, as demand for goods and unemployment is stabilised (even if
it’s for a short term). This will reduce the output gap and making it less
“expensive”[2] for the government to stabilise the economy with stimulus
spending. If a stimulus in a recession lacks short term stabilising effects
and/or took too long to implement, not only the output gap will be larger (thus
needing a larger stimulus), but it will have significant social cost (cost of
unemployment to households) and business investment will also take longer to
recover if decision makers in businesses are uncertain about future economic
prospects.

While the fiscal multiplier is important, its size should never be
the main determinant of whether a government is to implement a fiscal stimulus
or not. The most important determinant should always be whether if the economy
is in full employment. If President Obama does not want to use the 14th Amendment and the $1 trillion dollar platinum coin
option is out,
there needs to be a plan which the US government will need to enact to avoid further
austerity (when it is in fact, the austerity currently enacted by the government that is dragging down the economy) and implement stimulus. At the moment, the US government is failing
miserably to provide prosperity to its population despite the GFC, when employment to population ratio have no yet recovered to pre-crisis level for more than four years from the crisis.

[1] “Pay for itself” refers to a situation when an increase in
nominal public debt led to a reduction in public debt as a % of GDP. There are
examples of raw statistic correlation in the previous
post.

[2] “Expensive” only in a sense that if the government decides not
to print national currency to finance its spending. Neil Hart argues in his
paper (cited above) that the significance of the ability to print national
currency changes the whole debate of the “debt crisis” (if this actually exists
outside the European Monetary Union and other countries which do not have its
own national currency). In specific: